Exponential growth functions

We have dealt with linear functions earlier. All types of equations containing two unknown (x and y) variables may be inserted in a coordinate system. These types of equations are known as functions. A straight line is known as a linear function.

The function need not necessarily respond like a straight line equation. For example: If we have $50 000 deposited in the bank, and receive a 2 % interest annually, our investment shall increase as follows:

Year

Capital

Interest

Sum

1

50 000

50 000 · 0.02 = 1 000

51 000

2

51 000

51 000 · 0.02 = 1 020

52 020

3

52 020

52 020 · 0.02 = 1040.40

53 060.40

Compare that with what we would have with a linear increase (2%):

Year

Capital

Increase

Sum

1

50 000

50 000 · 0.02 = 1000

51 000

2

51 000

50 000 · 0.02 = 1000

52 000

3

52 000

50 000 · 0.02 = 1000

53 000

In this case we may note that the increase was constant each year. The investment may be described as:

Here we have an x-variable in the exponent. The interest and thus also the function are exponentials.

Now we shall examine the differences displayed with the functions in our example above in a coordinate system.

The lower straight line represents the linear increase and the upper bowed curve represents the exponential increase. In other words it is more profitable to have a compounded interest than a fixed return.

An exponential function is a nonlinear function that has the form of

$$y=ab^{x},\: \: where\: a\neq 0,b> 0$$

An exponential function with a > 0 and b > 1, like the one above, represents an exponential growth and the graph of an exponential growth function rises from left to right.

An exponential function where a > 0 and 0 < b < 1 represents an exponential decay and the graph of an exponential decay function falls from left to right.

When a quantity increases or decreases exponentially it increases or decreases by the same percent over equal time periods in comparison to when a compound increases or decreases linearly when a quantity increases or decreases with the same amount over equal time periods.